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Math 302
Modern Algebra
Spring 2009
Homework Assignments
Assignment 7
due Monday, April 6
4.3
Irreducibles and Unique Factorization
6.
Show that
x
2
+ 1 is irreducible in
Q
[
x
].
[
Hint:
Use a proof by contradiction. Suppose
x
2
+ 1 = (
ax
+
b
)(
cx
+
d
) for some
a, b, c, d
∈
Q
.
Show this leads to a contradiction.]
9.
Find all irreducible polynomials of
(b)
degree 3 in
Z
2
[
x
]. Be sure to indicate why each of these is in fact irreducible.
(c)
degree 2 in
Z
3
[
x
]. Be sure to indicate why each of these is in fact irreducible.
16.
Let
F
be a ±eld. Prove: For all
p
(
x
)
∈
F
[
x
],
p
(
x
) is irreducible in
F
[
x
] if and only if for every
g
(
x
)
∈
F
[
x
], either
p
(
x
)

g
(
x
) or
p
(
x
) and
g
(
x
) are relatively prime.
22. (a)
Show that
x
3
+
a
is reducible in
Z
3
[
x
] for each
a
∈
Z
3
.
4.4
Polyomial Functions, Roots, and Reducibility
3.
Use the Remainder Theorem to determine if
h
(
x
) is a factor of
f
(
x
).
(b)
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 Spring '03
 Edwards
 Algebra

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